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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A counterexample in nonlinear interpolation
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by Michael Cwikel PDF
Proc. Amer. Math. Soc. 62 (1977), 62-66 Request permission

Abstract:

If $({A_0},{A_1})$ is an interpolation pair with ${A_0} \subset {A_1}$ and T is a possibly nonlinear operator which maps ${A_0}$ into ${A_0}$ and ${A_1}$ into ${A_1}$ and satisfies ${\left \| {Ta} \right \|_{{A_0}}} \leqslant C{\left \| a \right \|_{{A_0}}}$ and ${\left \| {Tb - Tb’} \right \|_{{A_1}}} \leqslant C{\left \| {b - b’} \right \|_{{A_1}}}$ for all $a \in {A_0}$ and b, $b,b’ \in {A_1}$ and for some constant C, then it is known that T also maps the real interpolation spaces ${({A_0},{A_1})_{\theta ,p}}$ into themselves. We give an example showing that T need not map the complex interpolation spaces ${[{A_0},{A_1}]_\theta }$ into themselves. It is also seen that quasilinear operators may fail to preserve complex interpolation spaces.
References
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 62 (1977), 62-66
  • MSC: Primary 46M35; Secondary 47H99
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0433227-0
  • MathSciNet review: 0433227